as andreas said, obviously sometimes your board works, so you need to check if it works correctly then. best way to do that is to configure some fixed ip-address to rule out malfunctioning of your other computers, and try out different network-cables to rule out their malfunction. or alternatively andreas suggested to reconfigure the parallella linux image to output linux-messages to the serial port, so your log wont stop at the end of the boot-loader's output...
during the times it does not work, you said you get no serial output at all, so it breaks before even booting. not sure multimeter is the right tool for investigating then. you'd probably need to investigate how power changes in more fine-grained ways...